# Confidence Interval for 2 by 2 Odds

Odds = probability / (1 - probability) therefore odds can take on any value between 0 and infinity whereas probability may vary only between 0 and 1. Odds and log odds are therefore better suited than probability to some types of calculation.

Odds ratio (OR) is related to risk ratio (RR, relative risk):

RR = (a / (a+c)) / (b / (b+d))

When a is small in comparison to c and b is small in comparison to d (i.e. relatively small numbers of outcome positive observations or low prevalence) then c can be substituted for a+c and d can be substituted for d+b in the above. With a little rearrangement this gives the odds ratio (cross ratio, approximate relative risk):

OR = (a*d)/(b*c).

OR can therefore be related to RR by:

RR = 1/(BR+(1-BR)/OR)

..where BR is the baseline (control) response rate; BR can be estimated by b/(b+d) if not known from larger studies.

This function uses an exact method to construct confidence limits for the odds ratio of a fourfold table (Martin and Austin, 1991). The Fisher limits complement Fisher's exact test of independence in a fourfold table, for which one and two sided probabilities are provided here. Mid-P values are also given.

Please note that this method will take a long time with large numbers.

DATA INPUT:

Observed frequencies should be entered as a standard fourfold table:

 feature present feature absent outcome positive: a b outcome negative: c d

sample estimate of the odds ratio = (a*d)/(b*c)

Example

From Thomas (1971).

The following data look at the criminal convictions of twins in an attempt to investigate some of the hereditability of criminality.

 Monozygotic Dizygotic Convicted: 10 2 Not-convicted: 3 15

To analyse these data in StatsDirect select Odds Ratio Confidence Interval from the Exact Tests section of the analysis menu. Choose the default 95% two sided confidence interval.

For this example:

Confidence limits with 2.5% lower tail area and 2.5% upper tail area two sided:

Observed odds ratio = 25

Conditional maximum likelihood estimate of odds ratio = 21.305318

Exact Fisher 95% confidence interval = 2.753383 to 301.462338

Exact Fisher one sided P = 0.0005, two sided P = 0.0005

Exact mid-P 95% confidence interval = 3.379906 to 207.270568

Exact mid-P one sided P = 0.0002, two sided P = 0.0005

Here we can say with 95% confidence that one of a pair of identical twins who has a criminal conviction is between 2.75 and 301.5 times more likely than non-identical twins to have a convicted twin.